Conditional Probability Distributions in the Wide Sense1

Introduction. Doob [3, p. 29],2 has shown that if ()= [U, 11, u] is a probability space (i.e. 11 is a cr-field of subsets of a set U and u is a countably additive non-negative measure defined on 11, normalized by the condition u(U) = l) and 1l0 is a o--subfield of 11, and y is an re-dimensional random variable mapping U into an re-dimensional Euclidean space X, then y possesses a conditional distribution in the wide sense relative to llo. More specifically, he proved the existence of a function, p, of two variables F and w, where Y ranges over the Baire subsets of X, and w ranges over U, which satisfies the following three conditions: (i) p(Y, w) is, for each fixed w, a probability measure defined on the Baire subsets of X; (ii) p(Y, w) is, for each Y, a function measurable relative to llo; and (iii) JaP(Y, w)du(w) = u(A(~\y~l(Y)) for each A Gilo and each Baire subset Y of X. This result of Doob's is known to be false if one lets X be an arbitrary set with a distinguished cr-field of measurable subsets. It is our purpose (Theorem 1, below) to show that Doob's result does hold, at least if X is any locally compact, Hausdorff space whose one point compactification is metrizable. The idea behind this generalization is the following observation. If p(Y, w) exists with properties (i), (ii) and (iii) above then by (i) p(Y, w) determines for each fixed w, a unique element P(w) in the dual of C(X) where C(X) is the Banach space of all continuous realvalued functions on X which vanish at infinity. Furthermore from (ii) it follows that (g, P(w)) is for each g in C(X) a function measurable relative to 1lo where (g, P(w)) is the value of the linear functional P(w) at the vector g in C(X). And from (iii) it follows that Ja(&> P(w))du(w) =fAg(y(w))du(w) ior each AE^o and each g in C(X). This last is equivalent to JA(g, P(w))du(w) =/^(g, y(w))du(w) ior each A Gilo and each g in C(X) where for each fixed w in U, y(w) is the probability measure which lives at the one point y(w). Thus P is nothing other than the conditional expectation of y relative to 1lo. Here P and y are Banach space valued random variables. Thus if y possesses a conditional distribution in the wide sense relative to llo